7.3 - Polar Coordinates

Oct 13, 2026

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7.3 introduces polar coordinates as an alternative one-to-one mapping from points in the plane to ordered pairs (r, θ). The section defines radial and angular coordinates, relates them to Cartesian coordinates using right-triangle trigonometry, and discusses how restricting θ enables unique quadrant identification. It also presents conversion formulas between rectangular and polar systems, demonstrates worked examples, and explains the non-uniqueness and infinite representations of polar coordinates compared with rectangular coordinates.

LanguageEnglish

CategoryPaper Templates

FormatPDF

Pages20

Reading Time7 min

AccessFree to download

Learning Objectives

Locate points using polar coordinates

Convert between rectangular and polar coordinates

Sketch polar curves from equations

Convert equations between coordinate systems

Identify symmetry in polar curves and equations

Defining Polar Coordinates

Converting Points between Coordinate Systems

Example: Converting between Rectangular and Polar Coordinates

Non-uniqueness of Polar Representations

7.3 | Polar Coordinates Learning Objectives 7.3.1 Locate points in a plane by using polar coordinates. 7.3.2 Convert points between rectangular and polar coordinates. 7.3.3 Sketch polar curves from given equations. 7.3.4 Convert equations between rectangular and polar coordinates. 7.3.5 Identify symmetry in polar curves and equations. The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. This is called a one-to-one mapping from points in the plane to ordered pairs. The polar coordinate system provides an alternative method of mapping points to ordered pairs. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. Defining Polar Coordinates To find the coordinates of a point in the polar coordinate system, consider Figure 7.27 . The point P has Cartesian coordinates (x, y) . The line segment connecting the origin to the point P measures the distance from the origin to P and has length r. The angle between the positive x-axis and the line segment has measure θ . This observation suggests a natural correspondence between the coordinate pair (x, y) and the values r and θ . This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values (hence the term ordered pair) associated with it. In the polar coordinate system, each point also two values associated with it: r and θ . Figure 7.27 An arbitrary point in the Cartesian plane. Using right-triangle trigonometry, the following equations are true for the point P: cos θ = xr so x = r cos θ sin θ = yr so y = r sin θ . Furthermore, r 2 = x 2 + y 2 and tan θ = yx~~ ~~ . Each point (x, y) in the Cartesian coordinate system can therefore be represented as an ordered pair (r, θ) in the polar coordinate system. The first coordinate is called the radial coordinate and the second coordinate is called the angular coordinate. Every point in the plane can be represented in this form. Note that the equation tan θ = y/x has an infinite number of solutions for any ordered pair (x, y) . However, if we restrict the solutions to values between 0 and 2π then we can assign a unique solution to the quadrant in which the original point (x, y) is located. Then the corresponding value of r is positive, so r 2 = x 2 + y 2. This OpenStax book is available for free at [http://cnx.org/content/col11965/1.2](http://cnx.org/content/col11965/1.2) Theorem 7.4: Converting Points between Coordinate Systems Given a point P in the plane with Cartesian coordinates (x, y) and polar coordinates (r, θ), the following conversion formulas hold true: x = r cos θ and y = r sin θ, (7.7) r 2 = x 2 + y 2 and tan θ = yx~~ ~~ . (7.8) These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates. Example 7.10 Converting between Rectangular and Polar Coordinates Convert each of the following points into polar coordinates. a. (1, 1) b. (−3, 4) c. (0, 3) d. (53, −5) Convert each of the following points into rectangular coordinates. e. (3, π/3) f. (2, 3π/2) g. (6, −5π/6) Solution a. Use x = 1 and y = 1 in Equation 7.8: 2 2 2 r = x + y = 12 + 12 r = 2 tan θ = yx and = 11 = 1 θ = π 4 . Therefore this point can be represented as 2, π4 in polar coordinates. b. Use x = −3 and y = 4 in Equation 7.8: 2 2 2 r = x + y = (−3)2 + (4)2 r = 5 and tan θ = yx 4 = − 3 θ = −arctan⎛4 ⎞ ⎝ 3⎠≈ 2 .21. Therefore this point can be represented as (5, 2.21) in polar coordinates. c. Use x = 0 and y = 3 in Equation 7.8: 2 2 2 r = x + y = (3)2 + (0)2 = 9 + 0 r = 3 and tan θ = = y x 3 0 . Direct application of the second equation leads to division by zero. Graphing the point (0, 3) on the rectangular coordinate system reveals that the point is located on the positive y-axis. The angle between the p

FAQs

How do polar coordinates locate a point in the plane?

Polar coordinates represent a point using r (distance from the origin) and θ (angle from the positive x-axis). This provides an alternative mapping from points to ordered pairs.

What formulas convert between rectangular and polar coordinates?

The conversion uses x = r cos θ and y = r sin θ, along with r² = x² + y² and tan θ = y/x, which can be applied in either direction.

Why are polar coordinates not unique for a point?

A single point can correspond to infinitely many polar coordinate pairs because different angles (such as adding multiples of 2π) and even negative r values can produce the same x and y.

Oct 13, 2026

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